Questions tagged [young-tableaux]

For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.

Filter by
Sorted by
Tagged with
2 votes
1 answer
111 views

A problem about the existence of increasing coloring groups

Got stuck on this one for months. Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...
John Jiang's user avatar
  • 4,354
4 votes
0 answers
143 views

Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$: \begin{equation} d_\lambda = \sum_{a \in \mathrm{...
dmitry's user avatar
  • 133
5 votes
1 answer
218 views

Schur functors = Weyl functors in characteristic zero?

I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
Sunny Sood's user avatar
1 vote
1 answer
97 views

Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$

Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
Suzet's user avatar
  • 687
36 votes
1 answer
1k views

Errata for Fulton's "Young tableaux"

Fulton's Young tableaux is one of the best texts on the subject, one which I often recommend and cite for reference. Unlike Fulton/Lang and Fulton/Harris, it is neither an early-dawn draft nor a ...
5 votes
1 answer
111 views

geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety

For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...
staedtlerr's user avatar
2 votes
0 answers
74 views

Skewed plane partition with only row fillings reversed

The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
Zhi Wang's user avatar
-1 votes
1 answer
164 views

Orthogonality of irreducible and non-isomorphic representations [closed]

Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$. Does this ...
listener's user avatar
7 votes
2 answers
282 views

Decomposition of tensors into symmetry classes according to Schur functors

I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree. As it is well-known and extremely easy to ...
Bence Racskó's user avatar
1 vote
0 answers
125 views

Counting certain kinds of Semistandard Young Tableaux

We have a project in which it is natural to count the number of Young Tableau in which part of the weight has been specified. Does anybody know if this idea already appears in the literature? More ...
JRoss's user avatar
  • 270
3 votes
1 answer
252 views

Decomposition of tensor powers of the vector representation of $\frak{sl}_n$

Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$ V(\pi_1) \...
László Szabados's user avatar
3 votes
0 answers
115 views

Generalized Gaussian binomial and symmetric chain decomposition

Background Let $\mu = (\mu_1, \ldots, \mu_k)$ be a partition, meaning that $\mu_1 \geq \ldots \mu_k \geq 1$. The Young diagram associated to $\mu$ is given by the set $(r,c) \in \mathbb{N} \times \...
eti902's user avatar
  • 835
1 vote
1 answer
227 views

hook length formula for plane partitions

The hook length formula give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ...
Roger Van Peski's user avatar
8 votes
0 answers
215 views

Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
babu_babu's user avatar
  • 229
7 votes
0 answers
440 views

Mistakes in Logan and Shepp's famous paper on Young Tableaux?

In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
Matteo's user avatar
  • 106
3 votes
1 answer
525 views

References for applications of Young diagrams/tableaux to Quantum Mechanics

I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book: Wybourne, B.G.; "...
Malkoun's user avatar
  • 5,021
0 votes
0 answers
166 views

Young tableaux — irreps correspondence for simple complex Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally introduced to study the irreducible representations of finite symmetric groups $S_n$ ...
user267839's user avatar
  • 6,000
7 votes
1 answer
163 views

A formula for the generating function of Hoggatt binomials or of some Young tableaux

Let ${\left\langle\matrix {n \cr k}\right\rangle}_r$ denote the $r-$Hoggatt binomials defined by $${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \...
Johann Cigler's user avatar
7 votes
1 answer
261 views

Robinson-Schensted-Knuth (RSK) under restriction

I am curious about the following result concerning the Robinson-Schensted insertion procedure. I can formulate a proof via the Schützenberger evacuation operator, but I have struggled to find such an ...
fern-gossow's user avatar
3 votes
0 answers
136 views

Counting integer partitions below some Young diagram

Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
Yly's user avatar
  • 956
6 votes
0 answers
110 views

Bijection between forests and skew SYT + Cyclic sieving

Consider the two-row skew shape $\lambda_n = (2n+1,n)/(1)$. The number of standard Young tableaux of this shape is $\binom{3n}{n}-\binom{3n}{n-2}$ (since one can easily biject this to the set of non-...
Per Alexandersson's user avatar
5 votes
1 answer
258 views

On a proof involving Young symmetrizers acting on tensor spaces

I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
Bence Racskó's user avatar
1 vote
0 answers
126 views

$\mathfrak{sl}_2$-action on Young diagrams

Let $\mathcal{Y}$ be a vector $\mathbb{Q}$-space of all Young diagrams. Denote by $\delta_\lambda$ the Young diagram of the partition $\lambda$ and $c(\square)$ be the content of the square $\...
Leox's user avatar
  • 546
3 votes
0 answers
149 views

The Grassmann twist-map, an associated semi-group action, and RSK

Let me begin by setting some notation: Let $\mathrm{Mat}_{k,n}(\Bbb{R})$ denote the vector space of all $k \times n$ real-valued matrices. Given $g \in \mathrm{Mat}_{k,n}(\Bbb{R})$ and two (ordered) ...
Jeanne Scott's user avatar
  • 1,847
9 votes
0 answers
229 views

Hives for other root systems? [duplicate]

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
Igor Pak's user avatar
  • 16.3k
4 votes
0 answers
206 views

Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$

For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...
NicStr's user avatar
  • 59
0 votes
0 answers
89 views

Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
GGT's user avatar
  • 685
2 votes
0 answers
118 views

Yamanouchi ribbon tableaux?

Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions. The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ ...
eti902's user avatar
  • 835
1 vote
0 answers
129 views

Status of conjecture of Conrey and Gonek, combinatorial meaning

I was looking at the OEIS on the number of square Young Tableaux. In it Michael Somos referenced a paper of Conrey and Gonek, High Moment's of the Riemann Zeta-Function. Is there an combinatorial ...
yberman's user avatar
  • 771
1 vote
0 answers
71 views

Scalars by which symmetrizations of cyclic permutations act on Specht modules

Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$. Let $\...
Asav's user avatar
  • 163
1 vote
1 answer
186 views

Number of paths to a specific vertex in the Young's lattice

Consider the Young's lattice. What is the number of paths starting from the origin (0) to a specific Young diagram? For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 ...
TheTwistedSector's user avatar
2 votes
1 answer
131 views

Number of branches between two layers of the Young's lattice

In the Young's lattice, the number of branches that connect the $N$'th layer to the $N+1$'th layer has the sequence: $$ 1,2, 4, 7, 12, 19, 30, 45, 67, 97, 139, \cdots $$ Looking this up on OEIS, leads ...
TheTwistedSector's user avatar
12 votes
2 answers
366 views

Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
Zoltan Zimboras's user avatar
4 votes
0 answers
130 views

Map between irreducible representations in basis given by Young tableaux

Let $V$ be a $n$-dimensional complex vector space. Assume we have a $\mathbb C$-linear map $\varphi:\Gamma^{(a_1,\dots,a_n)}V\rightarrow \Gamma^{(b_1,\dots,b_n)}V$ between two irreducible ...
pi_1's user avatar
  • 1,433
1 vote
0 answers
68 views

LGV scheme: Any situations where the weights shift differently for each path?

In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
Thomas Kojar's user avatar
  • 4,449
9 votes
0 answers
480 views

Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
Sam Hopkins's user avatar
  • 22.9k
4 votes
2 answers
304 views

LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here). What do we mean by "time"? In the language of ...
Thomas Kojar's user avatar
  • 4,449
8 votes
0 answers
299 views

A question related to Young symmetrizers

Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
Abdelmalek Abdesselam's user avatar
2 votes
0 answers
95 views

The distribution of amajor over standard Young tableaux?

Given a standard Young tableau $T$, $i$ is called an ascent of $T$ if $i+1$ is in a higher row than $i$, and amajor$(T)$ is defined to be the sum of ascents of $T$. For the distribution of the amajor ...
xmchenhit's user avatar
  • 115
1 vote
0 answers
94 views

RSK correspondence for sum of two matrices

The celebrated RSK correspondence (see Wikipedia page) assigns to each integer $X$ matrix a pair of Young tableaux $P$ and $Q$. Now suppose that we have three integer matrices $X_1$, $X_2$, and $X_3$, ...
Sepehrius's user avatar
  • 121
9 votes
1 answer
919 views

A basic question about Young symmetrizers

This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I ...
Abdelmalek Abdesselam's user avatar
8 votes
0 answers
185 views

Generalized Young symmetrizers, why not?

For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$. We will denote by $G(\...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
122 views

A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation

Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
Sepehrius's user avatar
  • 121
8 votes
2 answers
719 views

Bender-Knuth involutions for symplectic (King) tableaux

First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
Sam Hopkins's user avatar
  • 22.9k
3 votes
0 answers
128 views

Consequences of Littlewood-Richardson rule

I am trying to read Deligne's paper 'Categories Tensorielles', and in the first chapter Deligne states some results obtained from the Littlewood-Richardson rule that I do not understand. He states: '...
S.Farr's user avatar
  • 275
3 votes
0 answers
77 views

Expansion of polytabloids in the standard basis

would like to know the most efficient way to write a polytabloid in terms of standard ones. I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
eti902's user avatar
  • 835
6 votes
1 answer
457 views

Refined reverse plane partition generating function

I have a simple question about the generating function for reverse plane partitions: $$\sum_{\pi \in RPP(\lambda)} z^{|\pi|}= \prod_{s \in \lambda} \frac{1}{1-z^{h_{\lambda}(s)}}$$ There's a ...
Samuel Crew's user avatar
8 votes
1 answer
475 views

RSK correspondence

Up to now, what are the difference ways we know to define RSK correspondence? I already know: By insertion and recording tableau. Ball construction or Viennot's geometric construction. Growth diagram ...
Mihawk's user avatar
  • 320
16 votes
3 answers
668 views

Plane partitions with equal margins

A plane partition of $n$ is an table of integers $A=(a_{ij})$ which add up to $n$ and non-increase in rows and columns. For example, $$A= \begin{matrix} 331 \\ 32 \ \ \\ 11 \ \ \end{matrix} $$ is a ...
Igor Pak's user avatar
  • 16.3k
3 votes
0 answers
123 views

Tableaux switching

I'm reading the article Tableau Switching: Algorithms and Applications by Benkart, Sottile, and Stroomer. Do you know if there are any articles or books that talk more about the properties of tableau ...
Mihawk's user avatar
  • 320